Let I ( f ) = <msubsup> &#x222B;<!-- ∫ --> a b </msubsup> f

Merati4tmjn

Merati4tmjn

Answered question

2022-05-09

Let I ( f ) = a b f ( x ) d x . The midpoint rule (open Newton-Cotes for n = 0) is I0(f)=(b−a)f(a+b2) I 0 ( f ) = ( b a ) f ( a + b 2 )
Show: For f C 1 ( [ a , b ] ) holds | I ( f ) I 0 ( f ) | ( b a ) 2 4 f

Answer & Explanation

charringpq49u

charringpq49u

Beginner2022-05-10Added 23 answers

Step 1
Define interval length h = b a and midpoint c = ( a + b ) / 2 .
Note that b c = c a = ( b a ) / 2 = h / 2. .
The midpoint approximation error is
E M = a b f ( x ) d x h f ( c ) = a b [ f ( x ) f ( c ) ] d x .
Using a Taylor approximation (or MVT), there exists ξ x between x and c such that
f ( x ) = f ( c ) + f ( ξ x ) ( x c )
We get
| E M | = | a b f ( ξ x ) ( x c ) d x | a b | f ( ξ x ) | | x c | d x f a b | x c | d x = h 2 4 f
Note that
a b | x c | d x = c b ( x c ) d x + a c ( c x ) d x = 1 2 ( b c ) 2 + 1 2 ( c a ) 2 = 1 2 h 2 4 + 1 2 h 2 4 = h 2 4

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